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\begin{document}
\title{Trade Flows, Multilateral Resistance, and Firm Heterogeneity\footnote{The authors wish to thank Gordon Hanson and two anonymous referees for comments and suggestions which greatly improved this paper. The authors also wish to thank James Anderson, Jeffrey Bergstrand, Elhanan Helpman, Peter Neary and
Adrian Wood, as well as participants at multiple seminars, for fruitful discussions. \textit{Disclaimer}: The views
expressed herein are those of the authors and should not be attributed to the IMF, its Executive Board, or its management, or to the Bank of England, the Monetary Policy Committee, or the Financial Policy Committee.}}
\author{Alberto Behar\footnote{International Monetary Fund. Email: ABehar@imf.org. Part of this work was conducted while at the Oxford Institute for Global Economic Development, Department of Economics, University of Oxford, and funding from the Economic
and Social Research Council and the World Bank are gratefully acknowledged.} \ \ \ \ \ \ \ \ Benjamin D. Nelson\footnote{Bank of England. Email: benjamin.nelson@bankofengland.co.uk.}
}
\date{
%January 2013
}
\maketitle

\begin{abstract}
% From accepted version:
% ====================
%Anderson and van Wincoop (2003) showed the importance of `multilateral
%resistance' general equilibrium effects in accounting for the response of
%trade flows to trade liberalisations. No studies have explored these effects
%in Helpman, Melitz and Rubinstein's (2008) extension of the Anderson and van
%Wincoop framework, which allows for firm heterogeneity, asymmetric trade
%costs, and zeros in trade flows. We do this by developing Taylor-approximated
%multilateral resistance terms with which to capture the general equilibrium
%effects of several comparative statics. These shed light on the relative
%quantitative significance of the different margins of adjustment of trade
%flows to changes in trade frictions. First, we show that for most bilateral
%trade cost changes, the net (general equilibrium) elasticity of trade is very
%close to the gross (partial equilibrium) effect, which itself varies with
%country size, and around half of which is due to firm entry. Second, most
%bilateral trade elasticities are positive for multilateral trade cost changes;
%that is, surprisingly, multilateral trade cost reductions tend to reduce
%bilateral trade flows because of strong multilateral resistance effects.
%Third, aggregated bilateral trade elasticities reveal a positive world-wide
%trade response to lower trade costs that is amplified slightly by firm entry
%but dampened by about two thirds because of general equilibrium effects.

% Shorter version
% =============
Anderson and van Wincoop (`AvW', 2003) showed the importance of `multilateral resistance' general equilibrium effects in estimating the response of trade flows to trade costs. We integrate this into Helpman, Melitz and Rubinstein's (2008) extension of the AvW framework, which allows for firm heterogeneity, in order to quantify the different margins of adjustment. For bilateral trade cost changes, the general equilibrium effects are small. Surprisingly, most country-pairs reduce their trade after a multilateral fall in trade costs. The global trade response to lower costs is positive but amplified by firm entry and significantly dampened by multilateral resistance.
\end{abstract}

\textit{JEL Classifications}: F10, F12, F14, F17 \

\textit{Key words}: Gravity models, multilateral resistance, firm
heterogeneity.

\section{Introduction}

How do changes in trade frictions affect trade flows? The answer to
this question is important for understanding the welfare implications of trade
liberalisations. This paper sheds light on this issue by examining comparative
statics in a gravity model that integrates two forces,\ namely multilateral
resistance (`MR') and firm heterogeneity, that were introduced in two
important papers in the literature.

In the first paper, \cite{anderson2003gravity}\ (`AvW') solve the
so-called `border puzzle' -- the implausibly large negative effect of the
US-Canadian border on trade between US states and Canadian provinces
highlighted by \cite{mccallum1995nbm}. AvW demonstrate that
traditional gravity equations capture the impact of only bilateral trade costs
on trade flows but ignore the fact that regions operate in a multilateral
world. As a result, traditional estimates fail to control for
theoretically-motivated price terms, which aggregate both domestic and
international trade costs and therefore capture MR. AvW show that bilateral
trade flows depend on bilateral trade costs \textit{relative to }MR. Failing
to account for MR typically leads one to overstate the importance of changes
in trade barriers on bilateral trade flows. Since then, further work has
studied the general equilibrium interplay between trade costs, trade flows,
and income.\footnote{
\cite{anderson2011terms} perform an analysis of the effects on the terms of trade -- and hence real
income -- of changes in trade costs arising from free trade agreements. \cite{behrens2009}
show the endogenous responses of wages and productivity, due to \cite{melitz2003iti}, matter for counterfactual trade flow analysis. They find that the estimated counterfactual border effect is around half the size of the `pure' border
effect ignoring these changes.}

The second paper on which we draw is
\cite{helpman2008estimating}\ (`HMR'). Heterogeneous firm productivity
within a country means not all firms are productive enough to cover the fixed
costs of exporting.\footnote{See \cite{bernard2007firms} for an overview
of firms in international trade.} If fixed costs are high enough, no firms in
a given country may find it profitable to export to a given destination.
Hence, in the presence of fixed costs of trade, `zeros' naturally arise in the
trade data --- a \textit{country} selection effect.\footnote{High fixed costs
of exporting from country $j$ to country $i$ do not imply the absence of
trade in the opposite direction, from $i$ to $j$. These asymmetric trade flows
are also a salient feature of the data that can be accommodated by modeling
firm heterogeneity.} In HMR's data, which we also use, the proportion of
countries that do not trade with each other or trade in only one direction is
around half of all observations. HMR explore a further implication. With
heterogeneous firm productivity and fixed costs of trade, a fall in variable
trade costs makes exporting firms export more, but also induces new firms to
export. These two effects are referred to as the \textit{intensive} and the
\textit{extensive} margins respectively. HMR argue that failure to account for
firm heterogeneity causes standard gravity estimation to conflate the impact
of trade costs on these two margins.

However, in performing their counterfactual exercise, HMR abstracted from the
general equilibrium effects of trade cost changes, and from the way in which these forces
interact with the intensive and extensive margins that those authors
distinguish. On the other hand, papers emphasizing MR have neglected the
implications of firm heterogeneity. Our contribution computes general
equilibrium comparative statics in a gravity model that marries the two
strands of the literature and facilitates implementation in large
cross-country datasets.

Our work is similar to \cite{eggeretal2011tte}, who demonstrate biases that arise from neglecting the general equilibrium
impact of changes in trade costs in the context of multiple Preferential Trade
Agreements. In that study, estimation addresses the issues of endogeneity,
country selection and MR, but only the last of these is also accounted for in
comparative statics.\footnote{The authors consider regressions that aim to
control for but do not separately identify firm heterogeneity.}

We present our hybrid AvW-HMR gravity model in section \ref{section_model},
which applies AvW general equilibrium closure to the extensive and intensive
margins of trade. In section \ref{section_frictionless}, we study the
theoretical comparative statics in a special case of our hybrid model -- a
frictionless initial equilibrium -- which helps us to illustrate the
mechanisms at work in the empirical comparative statics we subsequently
compute. Here we show that comparative statics will in general comprise three
terms: an intensive margin capturing the firm-level response to trade cost
changes, an extensive margin capturing the effects of firm entry, and an
adjustment term that captures the effects of MR. We show that changes in trade
flows at the extensive margin are also dampened by MR in a general equilibrium
setting. These comparative statics are useful for building intuition but are
inappropriate for empirical implementation. In particular, the frictionless
initial equilibrium is unrealistic and cannot generate the zeros and
asymmetries observed in the trade data.

Hence, section \ref{section_empirics} develops and empirically implements a
Taylor approximation for capturing comparative static MR effects. We adapt
Baier and Bergstrand's (2009) approach to a setting that includes firm
heterogeneity and accounts for zeros in trade flows, centering the
approximation at an initial equilibrium with positive asymmetric trade
frictions. Compared to AvW, Baier and Bergstrand (`BB') show that the
approximation error is small for the vast majority of country-pairs and we
believe the approach has a number of advantages over computational methods.
First, it preserves the possibility of analyzing bilaterally asymmetric trade
costs, which can explain asymmetric trade flows. In such settings, Bergstrand,
Egger \& Larch (2012) show that BB's flexible method is competitive with a
computational approach (like AvW's) that assumes bilateral symmetry in trade
costs. Second, this method does not require an estimate of the elasticity of
substitution between product varieties, which has a large effect on general
equilibrium comparative static outcomes (Bergstrand, Egger \& Larch, 2012).
Third, BB's approach is useful for gaining intuition about the effects at work
when trade costs change. We show in a heterogeneous firms setting that
multilateral resistance can be represented by three terms that are linear in
trade costs, capturing world trade resistance, importer MR, and exporter MR.
Having an analytically tractable expression makes the mechanics behind the
comparative statics exercises transparent.

After replicating the estimation exercise in HMR, our counterfactual analysis
decomposes overall trade flow elasticities into effects operating at the
intensive margin, the extensive margin, and through multilateral resistance
effects. Section \ref{section_empirics} continues by illustrating the relative
quantitative importance of each effect by means of three novel comparative
static results.

First, we analyze the case of two countries changing their bilateral trade
costs. We find that firm heterogeneity will tend to raise trade elasticities,
while MR effects are typically small for most country pairs. Firm
heterogeneity together with fixed costs of exporting imply the presence of an
extensive margin of trade that amplifies country-level trade responses above
those found in standard models. MR works against this, but the effects are
muted. So, for practical empirical purposes, we find that ignoring MR is
relatively innocuous for the majority of country pairs when only two countries
change their trade costs in isolation. Furthermore, we show that bilateral
liberalizations will tend to generate smaller trade responses at the extensive
margin for larger exporting countries, as there is less scope for firm entry
to take place for these exporters, compared to smaller exporters, in response
to trade liberalisations. This means that smaller countries tend to enjoy
larger trade responses overall when trade costs change bilaterally. The
average response indicates that around half the trade flow response operates
through the extensive margin of trade.

Second, we turn to multilateral changes in trade costs by analyzing the case
where all countries reduce their international trade frictions. Just as for
bilateral trade cost reductions, the extensive margin raises trade
elasticities, but now the effects of MR are much more important. Because
multilateral liberalizations imply small changes in relative trade costs for
any given country pair, trade elasticities\ net of MR are much smaller.
Surprisingly, we show analytically that bilateral elasticities can become
negative and empirically that this is true for most country pairs. That is,
multilateral trade liberalization will tend to redirect output across
destinations, reducing exports to some locations but increasing trade with
others. The pattern of responses is such that trade between smaller countries
tends to become redirected to larger importers. The reason is that, while
bilateral trade becomes more attractive, actual trade flows depend on the
costs incurred in this trade relative to other destinations. Because bigger
importers are less affected by MR, multilateral changes imply many changes in
relative prices that tend to favour exporting to larger importers over smaller
importers.\footnote{This is related to the `constructed home bias' -- or the
disproportionate share of domestic trade -- of \cite{anderson2010changing}. Because a larger share of large countries' trade is domestic rather than international, changes in trade costs have a smaller effect on large countries' price indices. This home bias effect is distinct from the `home market effect' -- whereby changes in trade costs shift the location of production -- as studied in, inter alia, \cite{krugman1980scale}, \cite{helpmankrugman1985}, \cite{davis1998home} and \cite{hanson2004home}.}

Third, we study the response of world trade to multilateral changes in trade
costs by aggregating the bilateral elasticities that properly capture the
general equilibrium effects of MR. Our main finding here is that the dampening
effect of MR dominates the amplification generated at the extensive margin.
The elasticity net of general equilibrium considerations is around one third
of the size of the aggregate trade elasticity that ignores relative price
effects and less than 40\% of those implied by standard models. In this
empirical application, country entry makes no contribution to the increase in
world trade and the role of firm entry is modest. The key results are summarized in Section \ref{section_conclude}.

\section{The model\label{section_model}}

The basis of our study is the gravity model proposed by Anderson and van
Wincoop (`AvW', 2003) and extended by Helpman, Melitz and Rubinstein (`HMR',
2008) to include firm heterogeneity. The environment is one in which CES
consumers within a set of endowment economies demand differentiated products
produced by monopolistically competitive firms that differ according to their
unit costs. The crucial feature of the HMR gravity set up is the presence of
fixed costs of exporting. This generates selection into exporting by the
lowest cost firms only. In some cases, trade costs can be so high that no
firms in a given country export to a particular destination. This generates
both firm selection and country selection into trade and the latter explains
the presence of `zeros' in bilateral trade data.

For brevity, we do not repeat the derivation of HMR's gravity equation here.
We begin where HMR left off. Their gravity equation relates imports by country
$i$ from $N_{j}$ firms in country $j$ to variable `iceberg' trade costs
$t_{ij}$, the fixed total output of the importer $Y_{i}$, an index of inward
multilateral resistance in the importer $P_{i}$, and a selection term $V_{ij}$
which measures the proportion of firms in $j$ that actively export to country
$i$:%
\[
M_{ij}=\left(  \frac{c_{j}t_{ij}}{\alpha P_{i}}\right)  ^{1-\sigma}N_{j}%
Y_{i}V_{ij}.
\]
(cf. HMR equation (6)). In this equation, $\sigma$ denotes the elasticity of
substitution between varieties, $\alpha \equiv1-(1/\sigma)$ and $c_{j}$
captures unit labour costs in exporter $j$.\footnote{In AvW, the equivalent
term is their `small $p$'.} We follow AvW in solving for the endogenous unit
costs by closing the model through assuming trade balance for each country.
This says that the total output of each exporter must equal the sum of imports
across all importing destinations, $Y_{j}=%
%TCIMACRO{\dsum \nolimits_{i\in I_{j}}}%
%BeginExpansion
{\displaystyle \sum \nolimits_{i\in I_{j}}}
%EndExpansion
M_{ij}$, where set $I_{j}$ denotes the set of countries that import from
country $j$. Using trade balance, HMR's gravity equation is:%
\[
Y_{j}=\left(  \frac{c_{j}}{\alpha}\right)  ^{1-\sigma}N_{j}%
%TCIMACRO{\dsum \nolimits_{i\in I_{j}}}%
%BeginExpansion
{\displaystyle \sum \nolimits_{i\in I_{j}}}
%EndExpansion
\left(  \frac{t_{ij}}{P_{i}}\right)  ^{1-\sigma}Y_{i}V_{ij}\Leftrightarrow
\left(  \frac{c_{j}}{\alpha}\right)  ^{1-\sigma}N_{j}=\frac{Y_{j}}{%
%TCIMACRO{\dsum \nolimits_{i\in I_{j}}}%
%BeginExpansion
{\displaystyle \sum \nolimits_{i\in I_{j}}}
%EndExpansion
\left(  \frac{t_{ij}}{P_{i}}\right)  ^{1-\sigma}Y_{i}V_{ij}}%
\]
Thus solving for unit factor costs and substituting back into the gravity
equation yields\footnote{A full derivation is given in the technical appendix
to this paper.}%
\begin{equation}
M_{ij}=\frac{Y_{i}Y_{j}}{Y^{I_{j}}}\left(  \frac{t_{ij}}{P_{i}\widehat{P}_{j}%
}\right)  ^{1-\sigma}V_{ij},\label{M}%
\end{equation}
in which the indices of \textit{inward multilateral resistance} $P_{i}$ and
\textit{outward multilateral resistance} $\widehat{P}_{j}$ are:%
\begin{align}
P_{i}^{1-\sigma}  & =%
%TCIMACRO{\dsum \limits_{j\in J_{i}}}%
%BeginExpansion
{\displaystyle \sum \limits_{j\in J_{i}}}
%EndExpansion
\left(  \frac{t_{ij}}{\widehat{P}_{j}}\right)  ^{1-\sigma}s_{j}^{J_{i}}%
V_{ij},\label{P}\\
\widehat{P}_{j}^{1-\sigma}  & =%
%TCIMACRO{\dsum \limits_{i\in I_{j}}}%
%BeginExpansion
{\displaystyle \sum \limits_{i\in I_{j}}}
%EndExpansion
\left(  \frac{t_{ij}}{P_{i}}\right)  ^{1-\sigma}s_{i}^{I_{j}}V_{ij}%
.\label{Phat}%
\end{align}
We define $Y^{I_{j}}\equiv {\displaystyle \sum \nolimits_{i\in I_{j}}} Y_{i}$ as the total output of all importers from country $j$. It differs from
AvW's notation only to the extent that fixed costs of exporting mean that
country $j$ firms will not export to all destinations. Using this notation, we
similarly define $s_{i}^{I_{j}}\equiv Y_{i}/Y^{I_{j}}$, such that
$s_{i}^{I_{j}}$ is the output share of country $i $ as a fraction of the total
output of all countries that import from $j$, $I_{j}$. We define the set
$J_{i}$ analogously -- as the set of all exporters to country $i$. Following
our convention, we then define $s_{j}^{J_{i}}\equiv Y_{j}/Y^{J_{i}}$ as the
output of country $j$ as a share of the total output of all exporters to
country $i$, $J_{i}$.\footnote{Note that, as long as the sets of active
traders $\{J_{i},I_{j}\}$ are fixed -- as for the comparative statics we
consider (see below) -- GDP shares will also be constant. We therefore
abstract from effects that trade cost changes might have on aggregate output
shares, $\{s_{j}^{J_{i}},s_{i}^{I_{j}}\}$. See Baier and Bergstrand (2009) and
AvW for similar set-ups. Anderson and Yotov (2011) compute terms of trade
effects arising from trade liberalisation, which would have implications for
real income and therefore world GDP\ shares to the extent that these effects
are asymmetric.} Variable trade costs are weakly greater than unity between
countries ($t_{ij}\geq1$ for $i\neq j$) but are exactly unity for trade within
countries ($t_{ij}=1$ for $i=j$). We do not require that variable trade costs
are bilaterally symmetric, which is important for capturing asymmetric trade
flow patterns. Finally, the selection term $V_{ij}$ would reduce to a constant
in the absence of fixed costs of trade. In the presence of fixed costs,
variation in the extensive margin of trade will contribute directly to
variation in aggregate bilateral trade flows $M_{ij}$ and indirectly because
the MR terms are also a function of the extensive margin.

How is the extensive margin of trade determined? In HMR's set up, there exists
a unit cost level above which firms in $j$ find it too costly to export to
$i$, which we denote by $a_{ij}$. Imposing general equilibrium closure as we
do allows us to write this cost cut-off as a function of outputs, trade costs,
and MR terms. Interestingly, it takes a gravity-like form, namely:%
\begin{equation}
a_{ij}^{\sigma-1}=\frac{1-\alpha}{N_{j}}\frac{Y_{i}Y_{j}}{Y^{I_{j}}}\frac
{1}{f_{ij}}\left(  \frac{t_{ij}}{P_{i}\widehat{P}_{j}}\right)  ^{1-\sigma
},\label{aij}%
\end{equation}
where $f_{ij}$ are the fixed costs of exporting from country $j$ to country
$i$. Hence a consequence of requiring general equilibrium closure is that the
extensive margin can be expressed in a convenient form akin to equation
(\ref{M}) for bilateral exports. Intuitively, higher fixed costs of trade
reduce the cost cut-off, so fewer firms in $j$ export to $i$. In addition, the
cost level above which firms in $j$ cannot export to $i$ rises as the product
of the two countries' outputs rises, as bilateral trade costs fall, or as MR
rises. The fact that MR enters equation (\ref{aij}) just as it does for
equation (\ref{M}) highlights the fact that valid comparative statics on the
extensive margin require accounting for general equilibrium effects, just as
AvW emphasize for valid comparative statics at the aggregate level.

Following HMR, if firm unit costs $a$ are Pareto distributed with shape
parameter $k$ on the interval $a\in \lbrack a_{L},a_{H}]$, then the extensive
margin selection term $V_{ij}$ entering equation (\ref{M}) can be written%
\begin{equation}
V_{ij}\equiv \max \left \{  \overline{k}\left[  \left(  \frac{a_{ij}}{a_{L}%
}\right)  ^{k-\sigma+1}-1\right]  ,0\right \}  ,\label{V1}%
\end{equation}
where $\overline{k}\equiv k/(a_{H}^{k}-a_{L}^{k})$. HMR relate the cost
cut-off $a_{ij}$ to a latent variable $Z_{ij}$, which is the ratio of the
variable profits of the lowest cost firm $a_{L}$ in $j$ to the fixed costs of
exporting. It is related to the cost cut-off according to%
\begin{equation}
Z_{ij}=\left(  \frac{a_{ij}}{a_{L}}\right)  ^{\sigma-1}.\label{Zij}%
\end{equation}
In our general equilibrium setting, this latent variable is in turn related to
MR through equation (\ref{aij}). In particular, we can write%
\[
Z_{ij}=\widetilde{Z}_{ij}\left(  P_{i}\widehat{P}_{j}\right)  ^{\sigma
-1},\text{ \  \ }\widetilde{Z}_{ij}\equiv \frac{(1-\alpha)a_{L}^{1-\sigma}%
}{N_{j}}\frac{Y_{i}Y_{j}}{Y^{I_{j}}}\frac{t_{ij}^{1-\sigma}}{f_{ij}}.
\]
In turn, this allows us to write the selection term in (\ref{V1}) as%
\begin{equation}
V_{ij}=\max \left \{  \overline{k}\left[  \widetilde{Z}_{ij}^{\delta}\left(
P_{i}\widehat{P}_{j}\right)  ^{\delta \left(  \sigma-1\right)  }-1\right]
,0\right \}  ,\label{V2}%
\end{equation}
where $\delta \equiv(k-\sigma+1)/(\sigma-1).$ Equations (\ref{M}) and
(\ref{V2}) then make clear that the comparative static effects of trade costs
will operate through three channels. First, changes in variable trade costs
will affect bilateral trade through the intensive margin, captured by the
$t_{ij}$ term in (\ref{M}). Second, to the extent that changes in trade costs
also affect the cost of trading with alternative export destinations, the
multilateral resistances in (\ref{M}) will change, tending to oppose the
direct effects of changes in trade costs through $t_{ij}$. Third, changes in
variable trade costs will also affect the extensive margin of trade through
$V_{ij}$. This final effect is also subject to competing forces. It comprises
both direct effects of changes in trade costs through the $t_{ij}$ term in the
numerator of equation (\ref{aij}), which by (\ref{Zij}) will affect the
extensive margin in (\ref{V2}), together with indirect effects through the MR
terms that enter (\ref{aij}). As in the aggregate trade flow equation
(\ref{M}), the MR effects operating through the extensive margin will tend to
oppose the direct effects of trade costs through the extensive margin,
providing some dampening effect. Overall, then, valid comparative statics must
account for these two sources of effects of MR on trade flows.

The log of equation (\ref{M}) gives the log-gravity equation:%
\begin{equation}
m_{ij}=y_{i}+y_{j}-y^{I_{j}}-(\sigma-1)\ln t_{ij}+w_{ij}+(\sigma-1)\ln
P_{i}\widehat{P}_{j},
\end{equation}
where lower cases denote logs and where we follow HMR in capturing $\ln
V_{ij}$ by
\begin{equation}
w_{ij}\equiv \ln \left(  \exp \delta z_{ij}-1\right)  ,\text{ \  \ }z_{ij}%
\equiv \widetilde{z}_{ij}+\ln \left(  P_{i}\widehat{P}_{j}\right)  ^{\sigma-1}.
\end{equation}
We denote the extensive margin elasticity by $\varphi_{ij}\equiv \partial
w_{ij}/\partial z_{ij}=\delta e^{\delta z_{ij}}/(e^{\delta z_{ij}}-1)$, which
captures the effects of changes in factors determining the latent variable
$z_{ij}$ on the proportion of firms exporting. Since we will be interested in
the effects of changes in trade costs on countries of different sizes, it is
useful to state the following result:

\begin{lemma}
\label{Lemma_grosssize}For given multilateral resistance, the derivative of
the bilateral elasticity of trade at the extensive margin $\varphi_{ij}$ with
respect to country size $s_{h}$, $h=i,j$, satisfies%
\[
\frac{\partial \varphi_{ij}}{\partial s_{h}}<0,\text{ \  \ }h=i,j.
\]
such that larger traders have smaller elasticities of bilateral trade at the
extensive margin with respect to trade costs.
\end{lemma}

\begin{proof}
See appendix.
\end{proof}

This result highlights one source of heterogeneity across countries' responses
to changes in trade frictions, namely, the extensive margin of trade. The
smaller the exporter, the greater the extent to which firm entry is encouraged
following a trade liberalisation. Intuitively, smaller, more remote exporters
have more scope for expanding the number of exporting firms than large,
well-connected traders. So the effect is more muted for larger exporters. The
extensive margin is not the only source of heterogeneity in countries'
responses to changes in trade costs, however. As we show next, multilateral
resistance also materially affects the response of trade flows to trade liberalisations.

\section{Comparative statics in a frictionless
world\label{section_frictionless}}

Before we turn to the empirical implementation of our model, we study its
properties in a simplified setting -- around a frictionless initial
equilibrium. At the end of the last section, we showed how country size drives
heterogeneity in trade responses at the extensive margin. In this section, we
illustrate how country size drives heterogeneous responses of trade flows
through multilateral resistance. The frictionless case is useful to illustrate
these general equilibrium effects in a way that abstracts from heterogeneity
at the extensive margin.

The thought experiment we perform is identical to one considered by AvW in
their framework, which abstracts from firm heterogeneity. To draw a comparison
with their results, consider a frictionless world in which variable trade
costs $t_{ij}=1$ for all $i,j$ and in which all countries trade such that
$I_{j}=J_{i}.$ The support of $a$ is such that all firms in country $i$ export
and that $V_{ij}=1$ for all $i,j$ at the initial equilibrium. In this world,
importer and exporter multilateral resistances are symmetric and equal to
unity, as in AvW's example. Using equations (\ref{aij}) and (\ref{V1}) in the
equilibrium we describe, a change in trade costs implies%
\begin{equation}
\frac{dV_{ij}}{dt_{ij}}=-(\sigma-1)\varphi \left(  1-\frac{dP_{i}}{dt_{ij}%
}-\frac{d\widehat{P}_{j}}{dt_{ij}}\right)  ,\label{delV_delt}%
\end{equation}
A rise in bilateral variable trade costs would reduce the number of firms
exporting. But these effects are dampened to the extent that multilateral
trade costs also rise, as captured by the third term. As is the case in
standard gravity models at the intensive margin, a higher elasticity of
substitution between products implies a large impact of trade costs on trade
flows. In the case of equation (\ref{delV_delt}) however, the effect here is
at the extensive margin, or the number of firms exporting. This standard
effect is multiplied by $\varphi=\delta(1+\overline{k})$, which reflects the
degree of heterogeneity across firms. When this is large, the extent of firm
entry (or exit) following trade cost changes is also large. Using
(\ref{delV_delt}) in (\ref{M}), a general expression for the effect of a
change in trade costs on trade flows evaluated at the initial frictionless
equilibrium is then given by:%
\begin{equation}
\frac{d}{dt_{ij}}\left(  M_{ij}\frac{Y}{Y_{i}Y_{i}}\right)  =-(\sigma
-1)\left(  1+\varphi \right)  \left(  1-\frac{dP_{i}}{dt_{ij}}-\frac
{d\widehat{P}_{j}}{dt_{ij}}\right)  ,\label{delM_delt}%
\end{equation}
which we can use to understand the components of trade flow comparative
statics. Just as the expression for comparative statics at the extensive
margin in (\ref{delV_delt}), equation (\ref{delM_delt}) shows that comparative
statics on overall trade flows between exporter $j$ and importer $i$ will
comprise an intensive margin captured by $\sigma-1$, some amplification due to
firm exit at the extensive margin due to the factor $1+\varphi>1$, together
with some dampening due to multilateral resistance $1-dP_{i}/dt_{ij}%
-d\widehat{P}_{j}/dt_{ij}$. The last effect captures the general equilibrium
repercussions of changes in trade frictions on average importer resistance and
average exporter resistance. On the importer side, trade flows are dampened to
the extent that cheaper varieties may now be available from elsewhere (i.e.
from exporters other than $j$), while on the exporter side, dampening may
occur because alternative export destinations may now be relatively more
attractive (i.e. importers other than $i$).\footnote{While we emphasise the
general equilibrium effect as operating through $\widehat{P}_{j}$, this
exporter MR\ term may be thought of as capturing effects through factor costs,
which are endogenous. In particular, a fall in $\widehat{P}_{j}$ corresponds
to a rise in $c_{j}$. See the expression above equation (\ref{M}), which we
used to solve for and eliminate $c_{j}$.}

We can gain further insight by continuing to explore the MR adjustment term.
Using equation (\ref{delV_delt}), we totally differentiate the system of price
indices given by (\ref{P}) and (\ref{Phat}) with respect to variable trade
costs. Evaluating at the frictionless equilibrium yields:%
\begin{align}
dP_{i}  & =\sum_{j}s_{j}dt_{ij}-\sum_{j}s_{j}d\widehat{P}_{j},\\
d\widehat{P}_{j}  & =\sum_{i}s_{i}dt_{ij}-\sum_{i}s_{i}dP_{i}.
\end{align}
Combining these two expressions results in the total differential of the MR
terms:%
\begin{equation}
dP_{i}+d\widehat{P}_{j}=-\sum_{l}\sum_{h}s_{l}s_{h}dt_{lh}+\sum_{l}%
s_{l}dt_{lj}+\sum_{h}s_{h}dt_{ih}.\label{mr}%
\end{equation}
This equation summarizes the behavior of endogenously determined MR to
exogenous changes in trade costs. It says that a rise in average trade costs
across all import and export destinations tends to raise MR, stimulating
bilateral trade between $i$ and $j$ (second and third terms), but that the
average world trade resistance has also risen, tending to reduce bilateral
trade (first term).

Consider first bilateral changes in trade costs. For a bilateral change, let
$dt_{lh}=dt_{hl}=dt$ for $l,h=i,j$, or else let $dt_{lh}=0$. Then using
(\ref{delM_delt}), GDP-weighted exports change according to%
\begin{equation}
\frac{d}{dt}\left(  M_{ij}\frac{Y}{Y_{i}Y_{i}}\right)  =-(\sigma
-1)(1+\varphi)\left(  1+2s_{i}s_{j}-s_{i}-s_{j}\right)
.\label{nofrict_bilateral}%
\end{equation}
Note that firm heterogeneity provides an amplification factor through the term
$(1+\varphi)$ when trade costs change, tending to raise the elasticity of
trade with respect to trade costs relative to AvW's homogeneous firms case. We
can also see from this expression that, whenever $s_{i}<1/2$, larger countries
experience smaller trade elasticities under bilateral change in trade costs.
Moreover, the effects of MR will typically be small under bilateral changes in
trade costs, as the term $1+2s_{i}s_{j}-s_{i}-s_{j}$ will typically be close
to unity. Intuitively, since only one set of trade costs changes under
bilateral liberalization, relative price changes are captured fairly well by
absolute price changes, such that only a small adjustment due to MR is required.

Consider next the effect of a multilateral change in trade costs, as in AvW,
such that $dt_{ij}=dt>0$ for all $i\neq j$, and $dt_{ii}=0$ for all $i$. Using
equation (\ref{delM_delt}) and our expressions for the total differentials in
(\ref{delV_delt}) and (\ref{mr}), GDP-weighted exports change according to%
\begin{equation}
\frac{d}{dt}\left(  M_{ij}\frac{Y}{Y_{i}Y_{i}}\right)  =-(\sigma
-1)(1+\varphi)\left(  -\sum_{l}s_{l}^{2}+s_{j}+s_{i}\right)
.\label{nofrict_multiateral}%
\end{equation}
This corresponds precisely to AvW's system (see their equation (15)), with the
addition of a term reflecting firm heterogeneity, $\varphi$. Compared to their
model, the firm heterogeneity term means the trade elasticity will be larger
by a factor $1+\varphi>1$. But as in AvW's model, it is the case that MR
dampens the effects of multilateral trade cost changes, and by more for
smaller countries. As such, larger countries experience larger trade
elasticities under multilateral changes in trade costs. The sign of the net
effect of a trade liberalization in exports is ambiguous in this world, and
the source of this ambiguity is the effect of MR through the term $-\sum
_{l}s_{l}^{2}+s_{j}+s_{i}$. Small country pairs (for which $s_{i}+s_{j}$ is
small) may well have negative bilateral elasticities. In principle, firm
heterogeneity will magnify the impact of any negative elasticities arising
from MR.

The analysis around the frictionless equilibrium has introduced the intuition
for our main results. Firm heterogeneity introduces an additional factor
$\varphi>0$ into trade elasticities. For bilateral changes in trade costs, MR
effects could be minor but, for multilateral changes, MR effects could be
sufficiently strong to make some trade elasticities negative.

We want to measure these effects, but the problem with implementing these
theoretical expressions empirically is that the world is obviously far from
frictionless. To the contrary, trade costs are significant \cite{anderson2004tc}
. Moreover, although the expressions above incorporate firm heterogeneity, we
lose an important dimension of variation across countries because the
$\varphi$ term has no country-pair subscript. Finally, the expressions above
do not allow the zeros in trade flows to play a role in determining likely
trade responses. As is clear from (\ref{P}) and (\ref{Phat}), zeros in the
trade data imply that the indices of MR should be adjusted to account for the
sets of active traders. Concretely, this means that the output shares captured
by the $s_{j}$ terms in the expressions above should be computed with respect
to the sets of countries with which exporters actually trade when trade
elasticities are generated, rather than all countries in the world.

\section{Empirical comparative statics\label{section_empirics}}

For estimation purposes, the methodology developed in HMR deals with firm
heterogeneity and country fixed effects control for importer and exporter
multilateral resistance.\footnote{The use of exporter and importer fixed
effects in estimation is acknowledged by AvW, Baier \& Bergstrand (2009) and
\cite{feenstra2004advanced} to be the most reliable estimation method in this context.} We replicate HMR's
approach to estimation. For comparative statics however, we adapt Baier and
Bergstrand's (`BB', 2009) `Bonus vetus OLS' (BVOLS) approach method for
approximating MR effects to a heterogeneous firms setting. In an homogeneous
firms setting, BB accomplish this by taking a Taylor approximation for the
non-linear MR terms around a world of symmetric but positive trade frictions.
Their method allows us to expunge the endogenous components from the right
hand side of the MR terms, such that approximate comparative statics are
straightforward to implement.

AvW use a computational procedure to solve for MR\ explicitly and to compute
comparative statics. Despite awareness of this procedure and the first-order
consequences of MR, the approach of AvW has not been applied to cross-country
data sets typically used in empirical applications of the gravity
model.\footnote{To our knowledge, the most ambitious exercise is that of
\cite{eggeretal2011tte}, who allow for MR comparative statics and other
factors on a dataset of 128 countries. Other contributions are still limited
to modeling US and Canadian regions together with a limited set of additional
countries or an aggregate for the rest of the world (AvW;
\cite{anderson2010changing}; \cite{anderson2011terms}; \cite{behrens2009}). All these studies conduct counterfactuals on binary variables like borders
or free trade agreements and not the broader class of continuous trade cost
proxies. \cite{BNMforth} apply the methods developed here to the case of logistics.} This is largely due to the difficulty in implementation, which
requires a customized program. Our application would be especially complex
because the system we study includes asymmetric bilateral trade costs, firm
selection, and different sets of active traders for different trading country pairs.

Like BB, our primary motive is to present a practical approach to computing
comparative statics that has low barriers to implementation in empirical
gravity work. It successfully corrects for the first-order inaccuracies
associated with the general equilibrium effects of trade cost changes.
\cite{baier2009bonus} show that their approximation is good for the majority of region pairs that
they consider. In particular, comparative statics of economic integration
based on the Taylor method are within 10\% of the AvW approach for 83\% of
their pairs. \cite{bergstrand2011gravity} provide Monte Carlo evidence that the trade-flow comparative statics computed
using a Taylor approach are close to those generated using nonlinear solvers.
This is despite using a version of BVOLS for estimation that, while better for
comparative statics, starts with the handicap of being biased at the
estimation stage. Furthermore, the accuracy of comparative statics improves as
the number of countries increases. Therefore, using fixed effects rather than
standard OLS with MR terms in estimation while using BVOLS for comparative
statics in a large cross section of countries, as we do, would yield even
better performance.

Further, the approach does not require us to assume particular values for
structural parameters like the elasticity of substitution, $\sigma$.
\cite{bergstrand2011gravity}
show that comparative statics are very sensitive to the choice of this
elasticity parameter and that the AvW approach is inaccurate when the assumed
elasticity is different from the actual
elasticity.\footnote{\cite{hertel2007confident} discuss the wide range
of $\sigma$ estimates and the sensitivity of general equilibirum comparative
statics to the value of $\sigma$.} A final advantage of the BB method is its
intuitive appeal. This, combined with the frictionless case just considered,
allows us to be clearer about the mechanisms that lie behind the comparative
statics we produce.\footnote{Note that, following Baier \& \ Bergstrand (2009)
and most dynamic macroeconomic models, we will consider a first-order Taylor
approximation to the multilateral resistance terms. A first-order
approximation will allow us to obtain analytical solutions but rules out
interactions between distance and other trade costs in the MR terms.}

To apply BB's method to the case of firm heterogeneity, we make the following
assumption for the purposes of comparative statics:\footnote{The assumption is
not required for the purposes of estimation, so HMR's empirical procedure is
not affected.}

\textbf{Decomposability}: \textit{The extensive margin terms }$V_{ij}%
$\textit{\ entering the indices of multilateral resistance }$\{P_{i}%
,\widehat{P}_{j}\}$ \textit{are approximately }$V_{ij}\simeq Z_{ij}^{\delta
}\left(  P_{i}\widehat{P}_{j}\right)  ^{\delta(\sigma-1)}$\textit{.}%
\footnote{\label{decompos_fn}HMR\ make a similar assumption in their Appendix
II. They do so for $V_{ij}$ in the main gravity equation while we do so only
in the price index terms. Furthermore, they assume that the $ij$ component of
$V_{ij}$ is symmetric. Their assumption is that $V_{ij}=\left(  \phi_{ij}%
\phi_{i}\widehat{\phi}_{j}\right)  ^{1-\sigma}$, in which $\phi_{ij}=\phi
_{ji},$ while $\phi_{i}$ and $\widehat{\phi}_{j}$ are importer and exporter
specific effects. The assumption that $\phi_{ij}=\phi_{ji}$ precludes
asymmetric trade flows, and for that reason is rejected by the authors. By
contrast, we do not impose such symmetry. In particular, we allow for
$\phi_{ij}\neq \phi_{ji}$ around our centre. In our case, this implies it is
possible that $\widetilde{Z}_{ij}\neq \widetilde{Z}_{ji}$.}

Using this, a Taylor expansion of importer and exporter multilateral
resistances yields a tractable expression for trade resistance that comprises
three terms:
\begin{multline}
(\delta+1)\ln \left(  P_{i}\widehat{P}_{j}\right)  ^{\sigma-1}\simeq
-\overset{\text{World Trade Resistance}}{\overbrace{\sum_{l\in I_{j}}%
s_{l}^{I_{j}}\sum_{h\in J_{i}}s_{h}^{J_{i}}\left[  (\sigma-1)\ln t_{lh}%
-\delta \widetilde{z}_{lh}\right]  }}\\
+\underset{\text{Importer's MR}}{\underbrace{\sum_{h\in J_{i}}s_{h}^{J_{i}%
}\left[  (\sigma-1)\ln t_{ih}-\delta \widetilde{z}_{ih}\right]  }}%
+\underset{\text{Exporter's MR}}{\underbrace{\sum_{l\in I_{j}}s_{l}^{I_{j}%
}\left[  (\sigma-1)\ln t_{lj}-\delta \widetilde{z}_{lj}\right]  }%
},\label{MRapprox}%
\end{multline}
in which $\widetilde{z}_{ij}\equiv \ln \widetilde{Z}_{ij}$. This expression has
three terms which, by analogy with (\ref{mr}), can be straightforwardly
interpreted. First, to the extent that world trade resistance is high, MR
between the $i$-$j$ trading pair will be low, tending to discourage
international trade on average. Second, to the extent that importer $i$ faces
high trade costs in obtaining output from other exporters in the set $J_{i}$,
exporter $j$ will export more to $i$. Third, when exporter $j$ finds it costly
to export to other destinations in the set $I_{j}$, it will tend to export
more to $i$ instead, all else equal. Hence the expression above has a
straightforward interpretation that makes clear the impact of
\textit{relative} trade costs on bilateral trade flows.

Moreover,\ our expression for MR incorporates the effects of firm
heterogeneity in two respects. First, trade resistance occurring through the
extensive margin is captured by the presence of the $\widetilde{z}_{ij}$ terms
on the right hand side. Their presence makes clear, for example, that MR is
affected by the intensive and extensive margins. Together with our discussion
of the role of MR in the extensive margin in section 3, this indicates a
two-way interaction between MR and the margins of trade. Second, the relevant
average trade costs that constitute MR in a world of firm heterogeneity are
taken over the sets of active traders. This is clear from the fact that the
summation terms are taken over the sets $J_{i}$ and $I_{j}$. This will have
important implications for comparative statics as we will see below.

Our estimation procedure, which replicates that of HMR, involves a two-step
process whereby a first stage probit regression is estimated for the
probability that country $j$ exports to country $i$. This probability is then
used to construct (a) a control variable for the extensive margin, $w_{ij}$,
and (b) the Inverse Mills Ratio as an additional control for country selection
into trade. Both of these are used in the second stage regression, which takes
the form of an otherwise standard gravity equation in which fixed effects
control for MR in estimation. Since the procedure and data is exactly as in
HMR, we provide an overview of the data, method and selected regression output
in the online technical appendix.

We turn instead to the implementation of our comparative statics, for which we
need three things. First, we need an estimate of the `firm level' intensive
margin elasticity of trade flows to trade costs. We proxy trade costs with a
number of observable variables and, for the elasticity, we take an estimate of
the coefficient on log distance, $\gamma.$ Our analytical results will be in
terms of changes in variable trade costs, assuming fixed costs stay constant,
and we will interpret the empirical illustration in this way. However,
observable proxies included in the probit stage could affect both fixed and
variable costs. The empirical illustrations, which are computed for changes in
distance, could therefore reflect changes in both costs. The estimates for the
trade cost elasticities are $\hat{\gamma}_{p}=0.66$ in the probit model and
$\hat{\gamma}=0.799$ in the 2nd stage regression.\footnote{We have made no
theoretical distinction between the intensive margin $\gamma$ and $\gamma
_{p}=\frac{\partial \widehat{\bar{z}}_{ij}^{\ast}}{\partial d_{ij}}$, which is
the distance coefficient from the probit estimate. Our simulations take
account of this small difference.} Second, we need country-pair extensive
margin elasticities $\varphi_{ij}=\delta e^{\delta z_{ij}}/(e^{\delta z_{ij}%
}-1)$, the estimate of which we denote by $\widehat{\varphi}_{ij}$. In turn,
these terms contain estimates of $z_{ij}$, a linear function of the the
variables in the probit model, together with $\hat{\delta}=0.72$. The
$\varphi_{ij}$ terms clearly vary by country pair and they provide one source
of cross country variation in trade elasticities. Finally, to make the
adjustments for MR, we need country GDP shares defined relative to the
relevant sets of active traders, $\{s_{i}^{J_{i}},s_{j}^{I_{j}}\}$ for all
$i,j$. These are easily generated from the matrix of observed trade flows.

\subsection{Gross elasticities\label{gross}}

Throughout our application it will be interesting to compare \textit{gross
elasticities}, which do not account for MR, with \textit{net elasticities},
which do. Because gross elasticities ignore MR, they will be identical for
both bilateral and multilateral changes in trade costs. The gross elasticity
for country pair $i$-$j$ with respect to variable trade costs is:%
\begin{equation}
\xi_{ij}^{gross}=(\sigma-1)(1+\varphi_{ij})
\end{equation}
where $\xi_{ij}^{gross}\equiv-\left.  \partial m_{ij}/\partial t_{ij}%
\right \vert _{P_{i},\widehat{P}_{j}}$. For the empirical analogue of this
elasticity based on a 10\% fall in distance, $\widehat{\xi}_{ij}%
^{gross}=\widehat{\gamma}(1+\widehat{\varphi}_{ij})$, we replicate the results
from HMR in Table \ref{table_gross}. For reference, the first row includes the
linear estimate, while the second row produces results from the NLS estimate
derived from the Pareto assumption made in section 2. The mean overall
elasticity of $1.564$ implies the bilateral country-level effect is due
roughly in equal parts to the intensive and the extensive margins. The
estimate is higher than would be implied by the linear OLS estimate or by
estimates attempting to deal with country selection alone ($1.21$ -- see HMR).
This is important because it implies one should allow for firm heterogeneity
even if one is not interested in the decomposition of the overall gross effect
into different margins of adjustment.\footnote{This point on country-level
comparative statics is distinct from the emphasis in HMR on the estimate of
$\gamma$, which is lower when controlling for firm heterogeneity than when
using OLS.
\cite{baranga2009}
and \cite{belenkiy2009} discuss the robustness of the HMR result.}

\bigskip
{\centering
[Table \ref{table_gross} about here]

}
\bigskip

Cross country variation in the extensive margin elasticities drives variation
in the estimated gross elasticities, which have a standard deviation of
$0.289$. The variation is generated from variations in $\widehat{\bar{z}}%
_{ij}^{\ast}$, which is monotonically related to the proportion of exporting
firms. Small country pairs that are far apart will have a low probability of
trading, which implies a low proportion of exporting firms and hence plenty of
scope for firm entry after a reduction in trade costs, as shown earlier in
Lemma 1.

This result is confirmed empirically in Table \ref{table_gross}, where the
correlation between $\xi_{ij}^{gross}$ and $s_{i}+s_{j}$ is $-0.286$. One can
analogously show that $\partial \xi_{ij}^{gross}/\partial \ln t_{ij}>0$.$\ $For
example, the correlation between the elasticity and log distance is $0.049$.
In addition to NLS estimates derived from the Pareto assumption, HMR also
produce estimates derived from a polynomial function of $z_{ij}$. We include
these estimates to show that the implications for bilateral elasticities are
similar, albeit with a larger standard deviation.

These estimates consider only actively trading pairs, but, as is clear from
the probit estimates, a fall in distance raises the predicted probability of a
pair of countries trading. To translate this continuous effect on the
probability into a comparative static simulation of binary country entry, a
new trading pair is formed (expanding the set $J_{i}$)\ if the reduction in
distance results in the most productive firm in $j$ now being able to cover
the fixed costs of exporting to $i$. Empirically, this means that the
predicted value of $z_{ij}$ after a fall in distance, $\widehat{\bar{z}}%
_{ij}^{\prime \ast}$, is positive when it previously was negative. Our analysis
counts in how many cases pairs that were not previously trading now have
$\widehat{\bar{z}}_{ij}^{\prime \ast}>0$.

For the default $10\%$ fall in distance, not a single instance of country
entry exists. Generating a case of country entry required fall in log distance
of $0.31$, or $27\%$. This implies that the distribution of distance and other
observed and unobserved trade costs is such that it would take very large
changes in trade costs to generate new trading pairs. This is consistent with
the observation in HMR that very little of the increase in world trade
observed over time is due to the formation of new trading pairs. Similarly, we
considered a rise in trade costs, estimating that a rise in log distance of
$0.35$ would have no effect on country entry and a rise in log distance of
$0.4$ would cause only one trading pair to stop trading. The results imply
that we do not need to consider changes in the sets of traders when studying
the net elasticities for the baseline $10\%$ change in trade costs that we do next.

\subsection{Bilateral changes in trade costs}

Allowing trade costs to change bilaterally results in the following:%

\begin{implication}

\label{propB}(Bilateral changes in trade costs) With Taylor approximated
multilateral resistance\ terms, the elasticity of bilateral trade flows to
bilateral (B) changes in variable trade costs is given by%
\begin{equation}
\xi_{ij}^{B}=\left(  \sigma-1\right)  \left(  1+\varphi_{ij}\right)  \left(
1+s_{i}^{I_{j}}s_{j}^{J_{i}}+s_{j}^{I_{j}}s_{i}^{J_{i}}-s_{j}^{J_{i}}%
-s_{i}^{I_{j}}\right) \label{xiB}%
\end{equation}
where $\xi_{ij}^{B}\equiv-dm_{ij}/d\ln t_{ij}$ for country pair $i$-$j$.
Bilateral trade elasticities are always decreasing in country size for
bilateral changes in trade costs.%

\end{implication}



\begin{proof}
See appendix.
\end{proof}

This is the empirical counterpart of the frictionless (\ref{nofrict_bilateral}%
), which we implement by employing:%
\begin{equation}
\xi_{ij}^{B}=\gamma+\varphi_{ij}\gamma_{p}-\left(  1+\varphi_{ij}\right)
\frac{\gamma+\delta \gamma_{p}}{1+\delta}\left(  -s_{i}^{I_{j}}s_{j}^{J_{i}%
}-s_{j}^{I_{j}}s_{i}^{J_{i}}+s_{j}^{J_{i}}+s_{i}^{I_{j}}\right)
\text{,}\label{xiB_empirical}%
\end{equation}
using our estimated parameters. Table \ref{table_B} shows this, together with
an entry demonstrating MR in the linear case, which disregards firm
heterogeneity, and an entry for the gross elasticity from section \ref{gross},
which disregards MR. Because most countries are small, bilateral changes in
trade costs have small MR implications. In the linear case, on average, the
dampening effect of MR is only $-0.033$, so the average bilateral elasticity
is close to that implied by the linear estimate of $\gamma$. If we allow for
firm heterogeneity but no MR, the comparative static effect is substantially
larger. The average MR effect is also small when we include firm
heterogeneity, so net ($\widehat{\xi}_{ij}^{B}$) and gross ($\widehat{\xi
}_{ij}^{gross}$) elasticities are close together. As a result, the average
amplifying effect of accounting for firm heterogeneity is much stronger than
the dampening effect of MR when trade cost changes are bilateral.

\bigskip
{\centering
[Table \ref{table_B} about here]

}
\bigskip

It is intuitive for the average MR effect to be small since the average
country is small. Since the essence of MR is to account for GDP-weighted
changes in relative trade costs, bilateral changes between two typical
countries have small general equilibrium effects. However, there are some
exceptions. Table \ref{table_lowest} illustrates this with specific country
examples and includes the ratio $\widehat{\xi}_{ij}^{B}/\widehat{\xi}%
_{ij}^{gross}\approx1+s_{i}^{I_{j}}s_{j}^{J_{i}}+s_{j}^{I_{j}}s_{i}^{J_{i}%
}-s_{j}^{J_{i}}-s_{i}^{I_{j}} $. Tiny pairs like Mauritania and Togo have
$\widehat{\xi}_{ij}^{B}/\widehat{\xi}_{ij}^{gross}\simeq0.9999$. \ By
contrast, Japan and the USA comprised 45\% of world GDP in the 1986 data and
generate a ratio of $0.79$. This suggests that MR effects are not
insignificant for all countries, even for bilateral changes in trade costs.
Even though Mexico and Spain were the 10th and 11th biggest countries in the
world, they each had less than 2\% of world GDP, such that the $\widehat{\xi
}_{ij}^{B}/\widehat{\xi}_{ij}^{gross} $ ratio for this pair is still
sufficiently close to unity to suggest it is only \textquotedblleft
very\textquotedblright \ big countries for which MR matters for bilateral liberalizations.

\bigskip
{\centering
[Table \ref{table_lowest} about here]

}
\bigskip

However, our adjusting for actual trading partners implies lower net-to-gross
ratios are also generated by small (and remote) exporters if they have a
dominant importer. An extreme example in Table \ref{table_lowest} is Japan,
which makes up almost three-quarters of importer GDP from Bhutan, so that the
net-to-gross ratio is only $0.26$ for exports from the latter to the former.
The table also contains some other examples. Thus, for bilateral changes in
trade costs, MR effects are material for (a) trade between the world's largest
country pairs and (b) exports from small exporters with few export
destinations to the world's largest countries. But given the skewness in the
distribution of country incomes, most country pairs are small, and are
therefore not materially affected by MR.

Since both the MR effect and the elasticity at the extensive margin are
decreasing in country size, the impact of size on the trade elasticities is
unambiguous in the case of bilateral changes in trade costs. Small country
pairs will tend to experience larger elasticities, as demonstrated in
Implication \ref{propB}. Figure \ref{fig_netB} illustrates this negative
relationship in terms of country GDP shares $s_{i}+s_{j}$ and the correlation
of -0.506 is stronger than was the case for $\widehat{\xi}_{ij}^{gross}$.

\bigskip
{\centering
[Figure \ref{fig_netB} about here]

}
\bigskip

\subsection{Multilateral changes in trade costs}

When all countries reduce trade costs, we find that:%


\begin{implication}

\label{propM}(Multilateral changes in trade costs) With Taylor approximated
multilateral resistance\ terms, the elasticity of bilateral trade flows to
multilateral changes (M) in variable trade costs is given by%
\begin{equation}
\xi_{ij}^{M}=(\sigma-1)(1+\varphi_{ij})\left(  -%
%TCIMACRO{\dsum \limits_{l\in I_{j}}}%
%BeginExpansion
{\displaystyle \sum \limits_{l\in I_{j}}}
%EndExpansion
s_{l}^{I_{j}}s_{l}^{J_{i}}+s_{i}^{J_{i}}+s_{j}^{I_{j}}\right) \label{xiM}%
\end{equation}
where $\xi_{ij}^{M}\equiv-\sum_{l}\sum_{h}dm_{ij}/d\ln t_{lh}$, in which $d\ln
t_{lh}=d\ln t$ for all $l\neq h$ and $d\ln t_{lh}=0$ for $l=h$. Then:

(a) after accounting for effects through multilateral resistance, it is the
case that%
\begin{equation}
\xi_{ij}^{M}\gtrless0,
\end{equation}
such that the sign of the elasticity of bilateral trade is ambiguous;

(b) (i) if the extensive margin is held constant ($\varphi_{ij}=0$),
\begin{equation}
\frac{\left.  \partial \xi_{ij}^{M}\right \vert _{\varphi_{ij}=0}}{\partial
s_{h}}>0,\text{ \  \ }h=i,j
\end{equation}
when $s_{i}^{I_{j}}+s_{i}^{J_{i}}<1$ if $h=i$, and when $s_{j}^{I_{j}}%
+s_{j}^{J_{i}}<1$ if $h=j$, such that larger countries larger firm-level
responses to multilateral trade liberalizations;

(b) (ii) if the extensive margin is allowed to change ($\varphi_{ij}>0$),
\begin{equation}
\frac{\partial \xi_{ij}^{M}}{\partial s_{h}}\gtrless0,\text{ \  \ }h=i,j
\end{equation}
such that the relationship between country size and the country-level
bilateral export elasticity is ambiguous.%


\end{implication}



\begin{proof}
See appendix.
\end{proof}

The expression in (\ref{xiM}) is the empirical counterpart to
(\ref{nofrict_multiateral}) in the frictionless case. In contrast to the case
of bilateral changes in trade costs, the elasticity of bilateral trade flows
with respect to multilateral changes in trade costs can be either positive or
negative in theory. If it is positive, it is likely to be far below the gross
elasticity that one would compute ignoring MR. The theoretical origin of this
result lies in the endowment economy studied here and in AvW. Changes in trade
costs serve to reallocate output from one destination to another so that
multilateral trade cost changes can mean that exports become redirected from
one importer to another even if the bilateral friction between all country
pairs falls. A fall in bilateral trade can be explained by the overall
bilateral cost rising \textit{relative} to the cost of importing from or
exporting to alternative destinations.\footnote{In parallel, the worldwide
reduction in trade frictions is associated with a rise in factor costs, which
reduce demand. To see the interplay between MR\ and factor costs, consider
what happens to trade between $i$ and $j$ when trade barriers between $j$ and
a third country $h\neq i$ fall, but the bilateral barrier between $i$ and $j$
remains unchanged. This manifests itself as a fall in $j$'s exporter MR. At
the same time, higher demand from the third country bids up $j$'s factor
costs. The higher factor costs make $j $'s goods more expensive in $i$ and,
given the barriers between $i$ and $j$ remain constant, this lowers demand
from $i$. So, the fall in exporter MR makes supplying goods to $i$ less
attractive in relative terms, while the rise in factor costs reduces demand by
$i$. This general equilibrium channel can be strong enough to reduce bilateral
trade between $i$ and $j$ even when the $i$-$j$ trade barrier is not constant
and actually falls along with other `third country' trade frictions, as in a
multilateral liberalisation.}

Part (b) (i) of Implication \ref{M} replicates AvW's `Implication 1' in the
environment in which we Taylor approximate MR. It states that countries
exporting to larger importers have larger elasticities of bilateral trade when
multilateral trade liberalization takes place. The reason is that larger
countries typically trade a smaller fraction of their output internationally,
instead trading proportionately more domestically. This means that large
countries are less affected by MR, which in turn means that the dampening
effect of MR on trade elasticities is smaller for large countries. This
remains true in our case as long as the extensive margin does not respond to
trade liberalizations ($\varphi_{ij}=0$).

Part (b) (ii) of the Implication arises because Lemma \ref{Lemma_grosssize}
acts against part (b) (i) of the Implication when the extensive margin is in
operation. Larger countries have smaller gross elasticities due to the
extensive margin being less responsive, but also have smaller MR effects, so
the theoretical relationship between country size and elasticity breaks down.
Therefore, the sign of the elasticity and its correlation with country size is
an empirical matter, which requires implementation using
\begin{equation}
\xi_{ij}^{M}=\gamma+\varphi_{ij}\gamma_{p}-\left(  1+\varphi_{ij}\right)
\frac{\gamma+\delta \gamma_{p}}{1+\delta}\left(  1+\sum_{l\in I_{j}}%
s_{l}^{I_{j}}s_{l}^{J_{i}}-s_{i}^{J_{i}}-s_{j}^{I_{j}}\right)  \text{.}%
\label{xiM_empirical}%
\end{equation}
Table \ref{table_M} demonstrates the dramatic effects of accounting for MR
when \textit{all} countries reduce distance. Because international frictions
have fallen relative only to domestic frictions, one might expect an average
effect close to zero. Indeed, accounting for MR almost completely cancels the
trade effect calculated in the absence of MR -- whether in the linear case or
when combined with firm heterogeneity. For example, the average net elasticity
$\widehat{\xi}_{ij}^{M}$ is only $0.006$, which is substantially lower than
the average gross elasticity $\widehat{\xi}_{ij}^{gross}$. Moreover, there are
only about 2,300 positive elasticities out of approximately 11,000 trade
flows, so Implication \ref{propM} (a) is relevant
empirically.\footnote{Similarly, Egger et al (2011) find that the signing of
multiple preferential trade agreements leads many signatories to experience
some falls in bilateral trade as a result of third country effects. This
illustrates how the general equilibrium concepts in this literature are
related to the traditional notion of trade diversion \cite{viner50}. Moreover, the structural gravity model provides a very explicit
interpretation of trade diversion, and one can show analytically that trade
between the signatory to an agreement and all third parties falls.
Furthermore, \cite{beharcirera2011} show how country size influences the effect of PTAs through MR.}

\bigskip
{\centering
[Table \ref{table_M} about here]

}
\bigskip

The theoretical analysis introduced the opposing forces in the relationship
between country size and the elasticity: bigger pairs have bigger MR
multipliers but lower gross elasticities due to the extensive margin. For
multilateral changes, the correlation between $\widehat{\xi}_{ij}^{M}$ and
$s_{i}+s_{j}$ is $0.490$. This positive correlation, which is illustrated in
Figure \ref{figM}, implies that the effect due to MR is stronger than that due
to the extensive margin in the multilateral case.

\bigskip
{\centering
[Figure \ref{figM} about here]

}
\bigskip

The general equilibrium effects we are describing are so strong that most
country pairs reduce their bilateral trade after reductions in their frictions
and trade is redirected elsewhere. Given the endowment economy model studied
here, negative export elasticities with some destinations should have
offsetting positive elasticities with others. For consistency with theory,
therefore, each exporter must have \textit{at least one }import destination
with which it has a positive export response. Empirically, this happens for
every exporter in our sample, but G7 importers are responsible for half of the
positive elasticities. Putting it starkly, most countries trade a lot more
with the G7 and less with almost everyone else. After a reduction in trade
frictions, they reduce their exports to most smaller countries to be able to
expand their exports to a handful of big countries.

Moving beyond bilateral trade responses, we aggregate the bilateral
elasticities by exporter, weighting each bilateral elasticity by the volume of
exports. As should be the case after aggregating across all export
destinations, all countries see an aggregate increase in exports even after
accounting for MR. The mean aggregate elasticity is only $0.29$, while
one-third of our countries have elasticities of less than $0.1$, a small
fraction of the elasticity implied by OLS. Researchers who are interested in
the comparative static impacts of trade costs for a particular country are
being particularly mislead by estimates based on gravity model coefficients
that ignore general equilibrium.

Some studies use gravity models to simulate the effects of multilateral
reductions in trade barriers on global trade (eg
\cite{wilson2005assessing}). It is interesting to calculate similar
world-wide elasticities, which we report in Table \ref{table_agg}. Ignoring
both MR and firm heterogeneity, the implied worldwide impact would be the
linear estimate of $\gamma$ reported in the first row of the table. Using the
same estimates but allowing for MR yields a much lower global impact, as shown
in the second row. In the third row, we allow for the extensive margin of
trade to operate, such that aggregating across all bilateral elasticities
yields a world elasticity $\widehat{\xi}^{gross}$\ of 1.291. This is close to
the minimum bilateral elasticity presented in Table \ref{table_gross} because
the world's biggest economies, which have a large weight in the aggregate
elasticity, tend to have the lowest individual elasticities. For global
elasticities (unlike bilateral elasticities), the intensive margin accounts
for over $60\%$ of the trade increase and the gross elasticity is about $10\%$
higher than that implied by OLS. Recall that there was no effect on country
entry after a $10\%$ fall in distance.

\bigskip
{\centering
[Table \ref{table_agg} about here]

}
\bigskip

Once we account for MR effects, however, world-wide elasticities are much
lower. As reported in the forth row of Table \ref{table_agg}, aggregation
yields a world-wide net trade elasticity $\widehat{\xi}^{M}$\ of $0.467$,
which is barely a third of the gross elasticity, and less than $40\%$ of the
elasticity implied by OLS. We draw the same conclusions from an analogous
empirical implementation using \ instead the polynomial estimates. Therefore,
existing gravity-based simulations of the global trade implications of
multilateral reductions in trade barriers are seriously misleading.

In sum, Table \ref{table_agg} confirms that MR dramatically reduces the
responsiveness of world trade to a multilateral reduction in trade frictions.
However, this masks the reorientation taking place bilaterally. We saw that
most bilateral trade elasticities are negative despite the larger gross
elasticities generated by effects at the extensive margin. This dramatic
impact is pervasive but greater amongst smaller trading pairs and shows the
redirection of exports away from most export destinations towards the G7.

\section{Concluding remarks\label{section_conclude}}

We have presented a gravity model for which we have computed comparative
statics that account for the effects of both firm heterogeneity and
multilateral resistance. To do so, we imposed general equilibrium closure on
Helpman, Melitz and Rubinstein's (2008) gravity model, rendering it comparable
to Anderson and van Wincoop's (2003) system while preserving asymmetries in
trade frictions and zeroes in the trade data. Adapting Baier and Bergstrand's
approximate multilateral resistance (MR) terms to the case of firm
heterogeneity allowed us to compare the effects of bilateral and multilateral
trade liberalizations in a context that accounts for both the extensive margin
of trade and the importance of relative trade costs in determining equilibrium responses.

In general, the presence of firm heterogeneity tends to increase the effect of
trade cost changes on trade flows, and the amplification is larger for smaller
countries. Therefore, even if one is not interested in the margins of trade,
they need to be considered when estimating the overall trade response. The
effects of firm heterogeneity will typically be dampened by MR however. Except
for the largest countries, the dampening due to MR effects is small for
bilateral changes in trade costs. One implication is that most analyses of
trade agreements between two countries (and by extension a handful of small
countries) can ignore MR for practical purposes.

By contrast, large effects due to MR emerge after multilateral changes in
trade costs, for example a comprehensive new multilateral trade agreement, are
considered. The theoretical sign of the bilateral trade response is ambiguous
and we find empirically that most bilateral trade elasticities are negative.
Because large countries are less affected by MR, exports get reoriented from
small importers to large importers.\footnote{In a counterfactual analysis that lies in between the bilateral and multilateral cases studied here, \cite{BNMforth} study the effects of a reduction in exporter-specific trade costs. A unilateral improvement in logistics quality increases exports by more in larger countries, chiefly because they are less affected by multilateral resistance.}

A further implication is that the world trade response to a
multilateral liberalization is positive but less than 40\% of that implied by
standard approaches that ignore MR. Properly accounting for MR may contribute,
for example, to explanations of why dramatic worldwide transport technology
improvements did not make a commensurately large contribution to trade
increases over time (\cite{behar2011}).

\newpage
\begingroup
\renewcommand{\enotesize}{\normalsize}
\theendnotes
\endgroup

\newpage

\begin{landscape}
\begin{table}[h] \centering
\caption{Summary statistics for estimates of gross country-level elasticity
$\xi_{ij}^{gross}$}
\setlength{\extrarowheight}{12pt}
\begin{tabular}
[c]{ccccccc}\hline
\textit{Functional Form} & \textit{Mean} & \textit{Median} & \textit{Std.
dev.} & \textit{Maximum} & \textit{Minimum} & Correlation with size\\ \hline
Linear & 1.176 & 1.176 & - & - & - & -\\
Pareto & 1.564 & 1.468 & 0.289 & 3.777 & 1.283 & -0.286\\
Polynomial & 1.853 & 1.846 & 0.518 & 2.995 & 1.141 & -0.348\\ \hline \hline
\end{tabular}
\label{table_gross}%
\end{table}%

\newpage

\begin{table}[h] \centering
\caption{Decomposition of trade flow elasticity for a bilateral reduction in trade frictions.}
\setlength{\extrarowheight}{12pt}
\begin{tabular}
[c]{cccccc}\hline
& \textit{Intensive} & \textit{Mean extensive} & \textit{Country entry} &
\textit{Mean MR} & \textit{Mean total}\\
& \textit{margin} & \textit{margin} &  & \textit{effect} & \textit{elasticity}%
\\ \hline
\multicolumn{1}{l}{Linear MR} & 1.176 & - & - & -0.033 & 1.143\\
\multicolumn{1}{l}{Firm heterogeneity} & 0.799 & 0.765 & Negligible & - &
1.564\\
\multicolumn{1}{l}{Combination} & 0.799 & 0.765 & Negligible & -0.037 &
1.526\\ \hline \hline
\multicolumn{6}{l}{{\small Intensive margin is OLS estimate of }$\gamma
${\small \ in the linear model and second stage NLS estimate of }$\gamma
${\small \ otherwise.}}\\
\multicolumn{6}{l}{{\small Extensive margin for each pair is }$\widehat
{\varphi}_{ij}\hat{\gamma}_{p}${\small . Country entry based on pairs where
}$\widehat{\bar{z}}_{ij}^{\prime \ast}>0${\small \ and }$\widehat{\bar{z}}%
_{ij}^{\ast}\leq0${\small . }}\\
\multicolumn{6}{l}{{\small MR is }$-\left(  1+\widehat{\varphi}_{ij}\right)
\frac{\widehat{\gamma}+\hat{\delta}\widehat{\gamma}_{p}}{1+\hat{\delta}%
}\left(  -s_{i}^{I_{j}}s_{j}^{J_{i}}-s_{j}^{I_{j}}s_{i}^{J_{i}}+s_{j}^{J_{i}%
}+s_{i}^{I_{j}}\right)  ${\small \ or linear analogue for each pair.}}\\
\multicolumn{6}{l}{{\small Total is from eqn (\ref{xiB_empirical}) or linear
analogue.}}%
\end{tabular}
\label{table_B}%
\end{table}%

\newpage

\begin{table}[h] \centering
\caption{Bilateral elasticities for specific country pairs.}
\setlength{\extrarowheight}{12pt}
\begin{tabular}
[c]{ccccccc}\hline
\multicolumn{2}{c}{\textit{Countries}} & \multicolumn{2}{c}{\textit{GDP
shares}} & Gross & Net & \textit{Ratio}\\
\textit{Exporter} & \textit{Importer} & \textit{Exporter} & \textit{Importer}
& $\widehat{\xi}_{ij}^{gross}$ & $\widehat{\xi}_{ij}^{B}$ & $\widehat{\xi
}_{ij}^{B}/\widehat{\xi}_{ij}^{gross}$\\ \hline
Bhutan & Japan & 0.001\% & 72.34\% & 1.49 & 0.39 & 0.26\\
Eq. Guinea & USA & 56.10\% & 42.85\% & 1.50 & 0.64 & 0.43\\
Kiribati & USA & 46.78\% & 51.92\% & 1.65 & 0.85 & 0.52\\
Solomon Isl. & USA & 45.38\% & 53.58\% & 1.56 & 0.84 & 0.54\\
French Guiana & USA & 42.76\% & 56.73\% & 1.41 & 0.80 & 0.57\\
USA & Japan & 29.23\% & 15.74\% & 1.28 & 1.02 & 0.79\\
Mexico & Spain & 1.721\% & 1.717\% & 1.29 & 1.25 & 0.97\\
Mauritania & Togo & 0.004\% & 0.005\% & 2.04 & 2.04 & 0.9999\\ \hline \hline
\end{tabular}
\label{table_lowest}%
\end{table}%

\newpage

\begin{table}[h] \centering
\caption{Decomposition of trade flow elasticity for a multilateral reduction in trade frictions}
\setlength{\extrarowheight}{12pt}
\begin{tabular}
[c]{cccccc}\hline
& \textit{Intensive margin} & \textit{Mean extensive margin} & \textit{Country entry} &
\textit{Mean MR effect} & \textit{Mean total effect}\\ \hline
\multicolumn{1}{l}{Linear MR} & 1.176 & - & - & -1.162 & 0.014\\
\multicolumn{1}{l}{Firm heterogeneity} & 0.799 & 0.765 & Negligible & - &
1.564\\
\multicolumn{1}{l}{Combination} & 0.799 & 0.765 & Negligible & -1.558 &
0.006\\ \hline \hline
\multicolumn{6}{l}{{\small Intensive margin is OLS estimate of }$\gamma
${\small \ in the linear model and second stage NLS estimate of }$\gamma
${\small \ otherwise.}}\\
\multicolumn{6}{l}{{\small Extensive margin for each pair is }$\widehat
{\varphi}_{ij}\hat{\gamma}_{p}${\small . Country entry based on pairs where
}$\widehat{\bar{z}}_{ij}^{\prime \ast}>0${\small \ and }$\widehat{\bar{z}}%
_{ij}^{\ast}\leq0${\small . }}\\
\multicolumn{6}{l}{{\small MR is }$-\left(  1+\varphi_{ij}\right)
\frac{\gamma+\delta \gamma_{p}}{1+\delta}\left \{  1+\sum_{l\in I_{j}}%
s_{l}^{I_{j}}s_{l}^{J_{i}}-s_{i}^{J_{i}}-s_{j}^{I_{j}}\right \}  ${\small \ or
linear analogue for each pair.}}\\
\multicolumn{6}{l}{{\small Total is from eqn (\ref{xiM_empirical}) or linear
equivalent.}}%
\end{tabular}
\label{table_M}%
\end{table}%

\newpage

\begin{table}[h] \centering
\caption{Impact of firm heterogeneity and MR on world-wide elasticities}
\setlength{\extrarowheight}{12pt}
\begin{tabular}
[c]{lccccc}\hline
& \textit{Intensive} & \textit{Extensive} & \textit{Country } & \textit{MR} &
\textit{Total effect}\\
& \textit{margin} & \textit{margin} & \textit{entry} & \textit{effect} &
\\ \hline
Linear {\small (no MR or heterog.)} & 1.176 & - & - & - & 1.176\\
Linear {\small (MR but no heterog.)} & 1.176 &  & - & -1.112 & 0.064\\
Firm heterogeneity {\small (heterog. but no MR)} & 0.799 & 0.492 &
Negligible & - & 1.291\\
Combination {\small (heterog. \& MR)} & 0.799 & 0.492 & Negligible & -0.824 &
0.467\\
Combination{\small \ (heterog. \& MR - poly.)} & 0.862 & 0.425 & Negligible &
-0.826 & 0.462\\ \hline \hline
\multicolumn{6}{l}{{\small Entries are trade-weighted aggregates of bilateral
elasticities calculated in table \ref{table_M} (or polynomial equivalents).}}%
\end{tabular}
\label{table_agg}%
\end{table}%

\end{landscape}%

\newpage
\begin{figure}[h]
\centering
  \caption{Net elasticity ($\widehat{\xi}_{ij}^{B}$) and country-pair GDP share when bilateral distance falls in isolation.}
  \label{fig_netB}
  \includegraphics[width=15cm]{Figure1.eps}
\end{figure}

\newpage
\begin{figure}[h]
\centering
  \caption{Net elasticity ($\widehat{\xi}_{ij}^{M}$) and country-pair GDP share when all countries reduce frictions.}
  \label{figM}
  \includegraphics[width=15cm]{Figure2.eps}
\end{figure}

\newpage
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